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A144441 Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

%I A144441 #9 Mar 04 2022 01:28:04
%S A144441 1,1,1,1,14,1,1,83,83,1,1,432,1550,432,1,1,2181,19898,19898,2181,1,1,
%T A144441 10930,217887,523548,217887,10930,1,1,54679,2199237,10589795,10589795,
%U A144441 2199237,54679,1,1,273428,21203828,184722860,362147222,184722860,21203828,273428,1
%N A144441 Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
%H A144441 G. C. Greubel, <a href="/A144441/b144441.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A144441 T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
%F A144441 Sum_{k=1..n} T(n, k) = s(n), where s(n) = 2*(2*n-3)*s(n-1) + 4*s(n-2) with s(1) = 1, s(2) = 2.
%F A144441 From _G. C. Greubel_, Mar 03 2022: (Start)
%F A144441 T(n, n-k) = T(n, k).
%F A144441 T(n, 2) = (1/2)*(7*5^(n-2) - (2*n+1)).
%F A144441 T(n, 3) = (1/8)*(4*n^2 - 5 - 14*(10*n-3)*5^(n-3) + 355*9^(n-3)). (End)
%e A144441 Triangle begins as:
%e A144441   1;
%e A144441   1,      1;
%e A144441   1,     14,        1;
%e A144441   1,     83,       83,         1;
%e A144441   1,    432,     1550,       432,         1;
%e A144441   1,   2181,    19898,     19898,      2181,         1;
%e A144441   1,  10930,   217887,    523548,    217887,     10930,        1;
%e A144441   1,  54679,  2199237,  10589795,  10589795,   2199237,    54679,      1;
%e A144441   1, 273428, 21203828, 184722860, 362147222, 184722860, 21203828, 273428, 1;
%t A144441 T[n_, k_, m_, j_]:= T[n,k,m,j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m,j] + (m*(k-1)+1)*T[n-1,k,m,j] + j*T[n-2,k-1,m,j]];
%t A144441 Table[T[n,k,4,4], {n,15}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 03 2022 *)
%o A144441 (Sage)
%o A144441 def T(n,k,m,j):
%o A144441     if (k==1 or k==n): return 1
%o A144441     else: return (m*(n-k)+1)*T(n-1,k-1,m,j) + (m*(k-1)+1)*T(n-1,k,m,j) + j*T(n-2,k-1,m,j)
%o A144441 def A144441(n,k): return T(n,k,4,4)
%o A144441 flatten([[A144441(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 03 2022
%Y A144441 Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144442, A144443, A144444, A144445.
%K A144441 nonn,tabl
%O A144441 1,5
%A A144441 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 05 2008